Finite Group Databasebeta

← #14418169

Subgroups — #14418169

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754138C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3#11754120C211yesmaximal, characteristic, complemented
4C4×#6710C211yesmaximal, characteristic, complemented
5C2×C5×#805C211yesmaximal, characteristic, complemented
6C2×C3×#3937C211yesmaximal, characteristic, complemented
7#11748478C211yesmaximal, characteristic, complemented
8#11747107C214yesmaximal, complemented
9#11747106C214yesmaximal, complemented
10#10610711—31nomaximal, Hall
11C2×C5×#235C2211yescharacteristic
12C2×C3×#709C2211yescharacteristic
13C2×C3×C42×C5C2211yescharacteristic
14C2×#6710C2211yescharacteristic
15C2×#6710C2211yescharacteristic
16C2×C5×#208C2211yescharacteristic
17C2×C3×#681C2211yescharacteristic
18#91309C412yescomplemented
19#91307C412yescomplemented
20#91299C412yescomplemented
21#91298C412yescomplemented
22#91297C2214yes—
23#91288C2214yescomplemented
24D3×#681—22no—
25#91286—22no—
26#91284C2214yescomplemented
27#91283—22no—
28D5×#208—22no—
29#91087C2212yescomplemented
30C4×C3⋊#168C2212yescomplemented
31C5×#805C2212yescomplemented
32C5×#805C2212yescomplemented
33C3×#3937C2212yescomplemented
34C3×#3937C2212yescomplemented
35#90992C2214yescomplemented
36#78980—51nomaximal, Hall
37#66156—31no—
38#66153—31no—
39#66079—31no—
40C2×C5×#104—31no—
41#65413—31no—
42C2×#3937D311yescharacteristic, complemented
43C4×#681—31no—
44#65113—34no—
45#65112—34no—
46C22×C3×C4×C5—21no—
47C22×C3×C4×C5C2311yescharacteristic
48C2×C5×#67—21no—
49C5×#235—21no—
50C5×#235—21no—
51C5×#235C2×C412yes—
52C5×#235C2312yes—
53C2×C5×D12C2×C411yescharacteristic
54C2×C4×C5×D3—21no—
55C2×C5×Dic6C2×C411yescharacteristic
56C2×C3×C22⋊D5—21no—
57C3×#709—21no—
58C3×#709—21no—
59C3×#709C2×C412yes—
60C3×#709C2312yes—
61C2×C3×D20C2×C411yescharacteristic
62C2×C3×C4×D5—21no—
63C2×C3×Dic10C2×C411yescharacteristic
64C3×C42×C5—21no—
65C3×C42×C5C2311yescharacteristic
66C3×C42×C5C2311yescharacteristic
67#6710C2×C412yes—
68#6710C2×C412yes—
69#6710D412yes—
70#6710D412yes—
71C2×C3⋊#168C2312yes—
72C2×C3⋊#168C2312yes—
73C5×#208—21no—
74C5×#208C2312yes—
75C2×C5×C3⋊C8—21no—
76C3×#681—21no—
77C3×#681C2312yes—
78C2×C3×#168—21no—
79#6661C2×C414yescomplemented
80#6656C2×C414yescomplemented
81#6655C2×C414yescomplemented
82#6652C2×C414yescomplemented
83#6650—24no—
84#6649—24no—
85D3×#168—24no—
86D5×C3⋊C8—24no—
87#5822—51no—
88#5819—51no—
89#5745—51no—
90C2×C3×#104—51no—
91#5124—51no—
92C2×#805D511yescharacteristic, complemented
93C4×#208—51no—
94#4824—54no—
95#4823—54no—
96C2×C5×#41—31no—
97C2×#709C2×D311yescharacteristic
98C2×C5×M16—31no—
99C2×C42×C5C2×D311yescharacteristic
100#4097—32no—
101#4096—34no—
102#4095—32no—
103#4094—34no—
104#4094—62no—
105#4088—32no—
106#4078—32no—
107C2×#681—31no—
108C2×#681—31no—
109C2×#681—62no—
110C2×#681C2×D311yescharacteristic
111#4053—34no—
112D5×M16—62no—
113#4047—62no—
114C5×#104—32no—
115C5×#104—32no—
116#3954—34no—
117#3937C2×D312yescomplemented
118#3937C2×D312yescomplemented
119#3935—32no—
120C4×#168—32no—
121#2956—151noSylow
122C23×C3×C5#4111yescharacteristic
123C22×C5×D3—21no—
124C22×C3×D5—21no—
125C2×C3×C4×C5—21no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—42no—
128C2×C3×C4×C5C22×C411yescharacteristic
129C2×C3×C4×C5C22×C411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5#4111yescharacteristic
133C2×C3×C4×C5C2411yescharacteristic
134C5×#67—22no—
135C5×#67—22no—
136C2×C5×Dic3—21no—
137C5×D12—21no—
138C5×D12C22×C412yes—
139C4×C5×D3—22no—
140C4×C5×D3—22no—
141C5×Dic6—21no—
142C5×Dic6C22×C412yes—
143C3×C22⋊D5—22no—
144C3×C22⋊D5—22no—
145C2×C3×Dic5—21no—
146C3×D20—21no—
147C3×D20C22×C412yes—
148C3×C4×D5—22no—
149C3×C4×D5—22no—
150C3×Dic10—21no—
151C3×Dic10C22×C412yes—
152C3⋊#168C22×C414yes—
153C3⋊#168C2×D414yes—
154C3×#168—22no—
155C5×C3⋊C8—22no—
156C2×C3×#41—51no—
157C2×#235C2×D511yescharacteristic
158C2×C3×M16—51no—
159C2×C3×C42C2×D511yescharacteristic
160#951—52no—
161#950—54no—
162#949—52no—
163#948—54no—
164#948—102no—
165#942—52no—
166#932—52no—
167C2×#208—51no—
168C2×#208—51no—
169C2×#208—102no—
170C2×#208C2×D511yescharacteristic
171#907—54no—
172D3×M16—102no—
173#901—102no—
174C3×#104—52no—
175C3×#104—52no—
176#822—54no—
177#805C2×D512yescomplemented
178#805C2×D512yescomplemented
179#803—52no—
180C4×C3⋊C8—52no—
181C5×#41—32no—
182C5×#41—32no—
183C5×#41—61no—
184C5×#41—61no—
185C2×C5×Q8—31no—
186C2×C5×D4—31no—
187C2×C5×D4—61no—
188C22×C4×C5—21no—
189C22×C4×C5—61no—
190C22×C4×C5C22×D311yescharacteristic
191C2×C22⋊D5—21no—
192#709—21no—
193#709—21no—
194#709C4×D312yes—
195#709C22×D312yes—
196C2×D20C4×D311yescharacteristic
197C2×C4×D5—21no—
198C2×Dic10C4×D311yescharacteristic
199C5×M16—32no—
200C5×M16—61no—
201C2×C5×C8—61no—
202C42×C5—21no—
203C42×C5C22×D311yescharacteristic
204C42×C5C22×D311yescharacteristic
205#689—34no—
206#688—34no—
207#687—34no—
208#686—34no—
209#681—21no—
210#681—32no—
211#681—32no—
212#681—32no—
213#681—32no—
214#681—64no—
215#681C22×D312yes—
216C2×#168—21no—
217C2×#168—32no—
218C2×#168—32no—
219C2×#168—64no—
220#676—64no—
221C8×D5—64no—
222#575—151no—
223#568—151no—
224#568—151no—
225#444—154no—
226#443—154no—
227#430—151no—
228C2×#104—31no—
229C2×#104—51no—
230C4×M16—151no—
231C22×C3×C5—41no—
232C22×C3×C5C4×D411yescharacteristic
233C22×C3×C5C4×D411yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C2×C5×D3—22no—
236C2×C5×D3—81no—
237C2×C3×D5—22no—
238C2×C3×D5—81no—
239C3×C4×C5—22no—
240C3×C4×C5—41no—
241C3×C4×C5C4×D411yescharacteristic
242C3×C4×C5C4×D411yescharacteristic
243C3×C4×C5C23×C411yescharacteristic
244C3×C4×C5C22×D411yescharacteristic
245C3×Dic5—22no—
246C5×Dic3—22no—
247C3×#41—52no—
248C3×#41—52no—
249C3×#41—101no—
250C3×#41—101no—
251C2×C3×Q8—51no—
252C2×C3×D4—51no—
253C2×C3×D4—101no—
254C22×C3×C4—21no—
255C22×C3×C4—101no—
256C22×C3×C4C22×D511yescharacteristic
257C2×#67—21no—
258#235—21no—
259#235—21no—
260#235C4×D512yes—
261#235C22×D512yes—
262C2×D12C4×D511yescharacteristic
263C2×C4×D3—21no—
264C2×Dic6C4×D511yescharacteristic
265C3×M16—52no—
266C3×M16—101no—
267C2×C3×C8—101no—
268C3×C42—21no—
269C3×C42C22×D511yescharacteristic
270C3×C42C22×D511yescharacteristic
271#216—54no—
272#215—54no—
273#214—54no—
274#213—54no—
275#208—21no—
276#208—52no—
277#208—52no—
278#208—52no—
279#208—52no—
280#208—104no—
281#208C22×D512yes—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—52no—
284C2×C3⋊C8—52no—
285C2×C3⋊C8—104no—
286ΓL(1,𝔽25)—104no—
287D3×C8—104no—
288C23×C5—61no—
289C23×C5#23711yescharacteristic
290C22×D5—21no—
291C5×Q8—32no—
292C5×Q8—61no—
293C5×D4—32no—
294C5×D4—62no—
295C5×D4—62no—
296C5×D4—61no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—42no—
300C2×C4×C5—61no—
301C2×C4×C5—62no—
302C2×C4×C5—62no—
303C2×C4×C5C2×C4×D311yescharacteristic
304C2×C4×C5C2×C4×D311yescharacteristic
305C2×C4×C5D3×D411yescharacteristic
306C2×C4×C5D3×D411yescharacteristic
307C2×C4×C5#23711yescharacteristic
308C2×C4×C5C23×D311yescharacteristic
309C22⋊D5—22no—
310C22⋊D5—22no—
311C2×Dic5—21no—
312D20—21no—
313D20C2×C4×D312yes—
314C4×D5—22no—
315C4×D5—22no—
316Dic10—21no—
317Dic10C2×C4×D312yes—
318C5×C8—62no—
319#168—22no—
320#168—34no—
321#168—34no—
322C2×#41—31no—
323C2×#41—51no—
324#137—152no—
325#137—152no—
326AGL(1,ℤ/8)—152no—
327AGL(1,ℤ/8)—152no—
328#135—154no—
329#131—154no—
330#131—154no—
331#131—302no—
332#131—302no—
333C2×M16—31no—
334C2×M16—51no—
335C2×M16—151no—
336C2×M16—151no—
337C2×M16—302no—
338C2×M16—302no—
339C2×C42D3×D511yescharacteristic
340#108—154no—
341#104—32no—
342#104—32no—
343#104—52no—
344#104—52no—
345#97—152no—
346C4×C8—152no—
347C2×C3×C5—21no—
348C2×C3×C5—21no—
349C2×C3×C5—21no—
350C2×C3×C5C2×C4×D411yescharacteristic
351C3×D5—42no—
352C5×D3—42no—
353C23×C3—101no—
354C23×C3#71111yescharacteristic
355C22×D3—21no—
356C3×Q8—52no—
357C3×Q8—101no—
358C3×D4—52no—
359C3×D4—102no—
360C3×D4—102no—
361C3×D4—101no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—42no—
365C2×C3×C4—101no—
366C2×C3×C4—102no—
367C2×C3×C4—102no—
368C2×C3×C4C2×C4×D511yescharacteristic
369C2×C3×C4C2×C4×D511yescharacteristic
370C2×C3×C4D4×D511yescharacteristic
371C2×C3×C4D4×D511yescharacteristic
372C2×C3×C4#71111yescharacteristic
373C2×C3×C4C23×D511yescharacteristic
374#67—22no—
375#67—22no—
376C2×Dic3—21no—
377D12—21no—
378D12C2×C4×D512yes—
379C4×D3—22no—
380C4×D3—22no—
381Dic6—21no—
382Dic6C2×C4×D512yes—
383C3×C8—102no—
384C3⋊C8—22no—
385C3⋊C8—54no—
386C3⋊C8—54no—
387C22×C5—41no—
388C22×C5—62no—
389C22×C5—241no—
390C22×C5#67⋊C411yescharacteristic
391C22×C5#67⋊C411yescharacteristic
392C22×C5C2×#23711yescharacteristic
393C2×D5—22no—
394C2×D5—81no—
395C4×C5—22no—
396C4×C5—41no—
397C4×C5—62no—
398C4×C5#67⋊C411yescharacteristic
399C4×C5#67⋊C411yescharacteristic
400C4×C5C22×C4×D311yescharacteristic
401C4×C5C2×D3×D411yescharacteristic
402Dic5—22no—
403#41—32no—
404#41—32no—
405#41—52no—
406#41—52no—
407#41—61no—
408#41—61no—
409#41—101no—
410#41—101no—
411C2×Q8—31no—
412C2×Q8—51no—
413C2×D4—31no—
414C2×D4—51no—
415C2×D4—61no—
416C2×D4—101no—
417C22×C4—21no—
418C22×C4—61no—
419C22×C4—101no—
420C22×C4C2×D3×D511yescharacteristic
421Q16—154no—
422SD16—154no—
423SD16—154no—
424D8—154no—
425M16—32no—
426M16—52no—
427M16—61no—
428M16—101no—
429M16—152no—
430M16—152no—
431M16—152no—
432M16—152no—
433M16—304no—
434M16—304no—
435C2×C8—61no—
436C2×C8—101no—
437C2×C8—152no—
438C2×C8—152no—
439C2×C8—304no—
440C2×C8—304no—
441C42—21no—
442C42C2×D3×D511yescharacteristic
443C42C2×D3×D511yescharacteristic
444C3×C5#295611yescharacteristic, complemented, Hall
445C22×C3—41no—
446C22×C3—102no—
447C22×C3—401no—
448C22×C3#402711yescharacteristic
449C22×C3#402711yescharacteristic
450C22×C3C2×#71111yescharacteristic
451C2×D3—22no—
452C2×D3—81no—
453C3×C4—22no—
454C3×C4—41no—
455C3×C4—102no—
456C3×C4#402711yescharacteristic
457C3×C4#402711yescharacteristic
458C3×C4C22×C4×D511yescharacteristic
459C3×C4C2×D4×D511yescharacteristic
460Dic3—22no—
461C2×C5—21no—
462C2×C5—21no—
463C2×C5—21no—
464C2×C5—122no—
465C2×C5C2×#67⋊C411yescharacteristic
466D5—42no—
467C23—61no—
468C23—101no—
469C23#678011yescharacteristic
470Q8—32no—
471Q8—52no—
472Q8—61no—
473Q8—101no—
474D4—32no—
475D4—52no—
476D4—62no—
477D4—62no—
478D4—61no—
479D4—102no—
480D4—102no—
481D4—101no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—42no—
485C2×C4—61no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—101no—
489C2×C4—102no—
490C2×C4—102no—
491C2×C4C4×D3×D511yescharacteristic
492C2×C4C4×D3×D511yescharacteristic
493C2×C4#677711yescharacteristic
494C2×C4#677711yescharacteristic
495C2×C4#678011yescharacteristic
496C2×C4C22×D3×D511yescharacteristic
497C8—62no—
498C8—102no—
499C8—154no—
500C8—154no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—202no—
505C2×C3C2×#402711yescharacteristic
506D3—42no—
507C5#7898011yescharacteristic, complemented, Sylow
508C22—41no—
509C22—62no—
510C22—102no—
511C22—241no—
512C22—401no—
513C22#9144211yescharacteristic
514C22#9144211yescharacteristic
515C22C2×#678011yescharacteristic
516C4—22no—
517C4—41no—
518C4—62no—
519C4—102no—
520C4#9144211yescharacteristic
521C4#9144211yescharacteristic
522C4C2×C4×D3×D511yescharacteristic
523C4C2×#677711yescharacteristic
524C3#1061071111yescharacteristic, complemented, Sylow
525C2—21no—
526C2—21no—
527C2—21no—
528C2—122no—
529C2—202no—
530C2C2×#9144211yescharacteristic
subgroupG/H
centerC4C2×#6777
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC2×C3×C42×C5C22
socleC2×C3×C5C2×C4×D4
prime radical#11754138C2
cosocleC2×C3×C4×C5C24
hypercenterC2×C42D3×D5
hypocenterC3×C5#2956
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11754125C2semidirect
3. #11754120C2semidirect
4. C4×#6710C2semidirect
5. C2×C5×#805C2semidirect
6. C2×C3×#3937C2semidirect
7. #11748478C2semidirect
8. #11747107C2semidirect
9. #11747106C2semidirect
18. #91309C4semidirect
19. #91307C4semidirect
20. #91299C4semidirect
21. #91298C4semidirect
23. #91288C22semidirect
26. #91284C22semidirect
29. #91087C22semidirect
30. C4×C3⋊#168C22semidirect
31. C5×#805C22semidirect
32. C5×#805C22semidirect
33. C3×#3937C22semidirect
34. C3×#3937C22semidirect
35. #90992C22semidirect
42. C2×#3937D3semidirect
79. #6661C2×C4semidirect
80. #6656C2×C4semidirect
81. #6655C2×C4semidirect
82. #6652C2×C4semidirect
92. C2×#805D5semidirect
117. #3937C2×D3semidirect
118. #3937C2×D3semidirect
177. #805C2×D5semidirect
178. #805C2×D5semidirect
444. C3×C5#2956semidirect
507. C5#78980semidirect
524. C3#10610711semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117541382
#117541252
#117541202
C4×#67102
C2×C5×#8052
C2×C3×#39372
#117484782
#117471072
#117471062
#106107113
#789805